Can you draw a path visiting the numbers in order without crossing itself?

Question

The rules of this challenge come from a recent Reddit post and are restated here.

Rules

Given a numbered grid, the challenge is to draw a path through the points in the grid subject to the following constraints:

1. The path takes a straight line between points.
2. The path does not cross itself (except at points) nor retrace any segment.
3. The path must visit every point in numbered order, but it is allowed to visit intermediate points. More precisely, the sequence of numbers the path visits must contain $1,2,\dots,9$ as a subsequence (see example).

Example

As an example (taken from the Reddit post), the following is a valid solution path for this particular numbered grid.

The order the path visits the vertices is 1, 7, 2, 3, 4, 2, 9, 5, 3, 6, 7, 4, 6, 8, 3, 9, and the bold subsequence shows that every point is visited in numbered order. Try tracing the path on paper to see for yourself.

Challenge

Find a path as described for the following numbered grid.

[This puzzle does not rely on tricks so, for example, the points are intended to be in a perfectly aligned grid.]

Answer 1

1, 4, 7, 2, 6, 3, 4, 5, 1, 8, 6, 7, 3, 8, 4, 9

... seems to work.

(Brief) explanation of how I found it:

The first thing I noticed about the grid was that the paths 1–2–3 and 6–7–8 were closely entwined and difficult to access without crossing or retracing, and so at least one of the two had to be very circuitous. I initially tried possibilities for 1–2 which involved one or two steps and found that all that I could find prevented 7–8 from being possible. Trying 1–8–6–2 resulted in the same effect. So the path from 1 to 2 had to be very circuitous indeed, and likely one of 1–3–7–2 or 1–4–7–2. I analyzed the latter first because it would allow 7–4–8 later on, and found that 2–6–3 allowed the 6–7 connection to be direct. From there, I filled in the gaps to reach my final solution.

Nice puzzle!